Problem $71$ below presents a model describing the drag of a particle moving through a

fluid medium that is released from rest at time $t=0($same initial conditions$).$ Using

Newton's Second Law, you build a model of the form:

$m{x}^{}=mg-F_{\text{d}\text{r}\text{a}\text{g}},(governing$ equation$)$

$v(0)=x^{}(0)=0,(initial$ velocity$)$

where $x=x(t)$ is the particle's position, $m$ is the mass of the particle, $g$ is the acceleration

due to gravity, and $F_{\text{d}\text{r}\text{a}\text{g}}$ is the magnitude of the drag force. You account for drag using

the formula $F_{\text{d}\text{r}\text{a}\text{g}}=c_d{v}^{}$ for known values of drag coefficients. Here $=1$ for a very

viscous fluid.

Problem $71($Small Reynolds Number $($I$)).$ In the case of small Reynolds number

$(Re1),$ the drag force on an object moving through a fluid, denoted by $F_{\text{d}\text{r}\text{a}\text{g}},$ is known

to be directly proportional to the speed of the object relative to the fluid:

$F_{\text{d}\text{r}\text{a}\text{g}}=kv$

where $k$ is a dimensional constant of proportionality. Assume that a tiny particle is

released from rest at the origin and take the $y-$axis pointing downward in the direction of

motion with the origin at the initial position. In this problem you will be asked to derive

a formula for the equation governing the motion of the object and solve this equation

to answer the following questions. Be sure to verify that your formula is dimensional

consistent.

$[2$ pt$]($a$)$ Draw a free$-$body diagram for this situation and use your free$-$body diagram

with Newton's second law to write down the governing equation for this situation.

$[2$ pt$]($b$)$ Derive an expression for the velocity as a function of time $v=v(t).$

$[2$ pt$]($c$)$ Integrate the expression for $v=v(t)$ to find the position of the particle as a

function of time $y=y(t).$

$[2$ pt$]($d$)$ Determine the speed of the terminal velocity $v_{\text{t}\text{e}\text{r}\text{m}}.$

$[2$ pt$]($e$)$ Express the large $t$ behavior of the position function $y=y(t)$ in terms of $v_{\text{t}\text{e}\text{r}\text{m}}.$